09 February 2008

A mathematician goes to the post office

There's a pretty standard theorem that one proves at the beginning of, say, a differential geometry course -- that any (n-1)-dimensional manifold embedded in n-dimensional Euclidean space is locally the graph of some function. (To any geometers are reading this, I apologize if I haven't stated this correctly.)

This morning I had to go to the post office to pick up a piece of certified mail. I live on one side of a large set of railroad tracks; the post office is about a mile away on the other side of the tracks. I generally don't go past the tracks, because very little that interests me is there, and I hadn't looked at a map before setting out. It's important to know where the tracks are when walking around in that neighborhood, because there are a lot of streets that exist on both sides of the tracks but are interrupted by them. The street on which the post office is, Florence Avenue (links goes to Google satellite imagery) may be such a street; it exists on both sides of the tracks, and doesn't cross over the tracks. It might cross under; Google Maps won't tell me, and I'm not curious enough to head out there again and look. So the scheme of "follow the street with the same name" doesn't always work.

I knew the tracks run basically from northwest to southeast. (For those who know where I live, or who are looking at the map I linked to: all directions are "nominal", i. e. by "west" I mean "the direction in which the street numbers get higher".) At one point on the way back I crossed a bridge and I was wondering "could that have been the tracks I just walked over?" Then I thought "hmm... I'm further north than I was when I crossed the tracks heading to the post office, so the tracks must be further west here."

Why must they? Because the tracks are basically a one-dimensional manifold embedded in two-dimensional Euclidean space. So locally there's a function mapping east-west streets to north-south streets that gives the position of the tracks. Everyone knows this intuitively. If you asked someone who knows that neighborhood well "Where does [insert street here] cross the tracks?" you'd get an answer, even if they're totally ignorant of geometry.